First, a disclaimer: I don't think Palin has the necessary experience to be president. What she does have is a very appealing personal narrative, plus identity politics (female, rural, etc.). Apparently that same standard was not used in picking our president in 2008, as Obama is the other side of the same coin.
However, I do "like" her, for reasons not all of her own making. I like the fact that she has all the right enemies. The fact that her "red-state," rural, pro-life, pro-gun "normalcy" drives the beltway crowd nuts makes me chuckle. And this includes both the left and the snobs of the right, like David Brooks, etc.
I like her for reasons of sympathy. Her treatment at the hands of the media, particularly in light of the kid-gloves treatment of Obama (for example, 11 AP fact-checkers for her book, where the only thing fact-checked on him is whether SNL was too tough on him) both validates my prejudices about media bias and makes me want to defend her (even when she deserves the criticism, by the way). I felt the same way about Reagan, George W. Bush, Cheney, Dan Quayle, and even poor, stupid, and naked Carrie Prejean.
This leads to the "anti-intellectual" business. Many conservatives have gotten tired of being told for years that we, and those who agree with us, are stupid. First, Reagan was stupid (even though his voluminous writings in his own handwriting indicate otherwise), but Jimmy Carter, in spite of his feckless presidency was brilliant. Then was Quayle. Then Bush 43. Having a Yale BA and a Harvard MBA counted for nothing--he MUST be dumber than Gore or Kerry, even when test scores and transcripts proved them to be almost exactly on the same intellectual level. Meanwhile, we see plenty of Ivy-league types who seemingly couldn't run a lemonade stand. I work in academics, and I know these people--some of the dumbest geniuses you'll ever meet. So the visceral reaction against that level of prejudice is to perversely translate lack of demonstrated intellect into a badge of honor compared to the know-nothing "elites." We haven't forgotten that Truman didn't go to college, nor that Reagan went to Eureka. That doesn't make the prejudice any more correct, but it's there. And honestly, I think I'd rather see a President or Supreme Court Justice with top marks from the University of Tennessee than yet another Harvard-Yale-Princeton type, particularly if there are questions as to whether "diversity" may have influenced their admission. (If Bush's family explains why he got into Yale, can't we question whether Obama's unique heritage may have influenced his admission to Harvard, especially in the absence of a transcript?)
Anyway, all of that said, I wouldn't vote for her. But I understand why she lights people's fire (on both sides of the aisle). I'll probably get flamed for saying anything remotely complimentary about her, but that's an honest answer.